The texture is a flat image, usually. JPG that you can add/overlay to your photo with any software or app supporting layers and blending modes - PS CC, PSE, Gimp or Procreate app.
How to apply texture overlay the right way to achieve a fine art look?
1. Pick the photo, the right one. Yes, all should start with the correct photo, edited or not but the photo should have good light and tone of the background. Pick the solid grey, brown, or black background and try to avoid bright colors with high contrast (it will be hard to blend the texture).
2. Think about the look you like to get. There are many different textures - soft ones, floral ones, painted ones, grunge ones. What will work with one photo it won't do any better than other ones.
3. Learn basic knowledge of Photoshop blending modes and layers. Read from the adobe.com website
4. Be creative and don't be afraid to mix few textures and try different blending modes or editing tools (you can change the color of your texture, add more contrast, use gradient filters).